A note about Euler angles

The goal of any 3D reconstruction technique is, for each 2D image, to obtain 2 translational parameters, and 3 orientational parameters. For reference-based alignment., these angles are obtained by knowing the projection angles of a similar reference structure. In random conical tilt, 1 of the Euler angles is arbitrary, angle phi, the in-plane rotation angle, (the "random" aspect of random conical tilt). The last 2 Euler angles are derived directly from the geometry of the data collection: angle theta is the magnitude of the tilt, and angle psi is the direction of the tilt axis in the (tilted-specimen) image.

In the particle-windowing step windowparticles.rct, in addition to the outputs Zerodegr/2-untilted and Tilted/2-tilted, there are additional outputs, Zerodegr/stk0-unrot and Tilted/stk0-unrot, which correspond directly to the images windowed from the micrographs. The former pair of outputs have been rotated such that the tilt axis is vertical.

The reason for this step is that there is a potential degeneracy for how WEB and JWEB determine the angles; the angle theta and theta+180 would be equally correct. Under some conditions, this degeneracy could cause problems downstream. The batch file storeangles.rct will write the angles to the terminal, and the micrographs' displayed tilt parameters SHOULD be similar to each other. However, during the fitting of the angular parameters during tilt-pair picking, there is a possibility that equivalent (as far as the fitting is concerned) solutions are chosen inconsistently. Thus, in the reconstruction step, we will use the pre-rotated versions. The doc file containing Euler angles, dangles, from storeangles.rct will have angle psi set to 0, and this file will be ultimately used for the reconstruction. For example:

 ;rct/dat   09-OCT-2012 AT 17:49:47   Tilted/dangles.dat
 ;       ANGLE_PSI    ANGLE_THETA    ANGLE_PHI
    1 3   0.00000       45.3470       80.0500
    2 3   0.00000       45.3470       274.760
    3 3   0.00000       45.3470       272.790
    4 3   0.00000       45.3470       75.9200
    5 3   0.00000       45.3470       99.2100
    6 3   0.00000       45.3470       36.8600                                                                                          
[...]
 1581 3   0.00000       42.6240       219.010
 1582 3   0.00000       42.6240       161.240
 1583 3   0.00000       42.6240       228.220
 1584 3   0.00000       42.6240       207.330
 1585 3   0.00000       42.6240       342.630
 1586 3   0.00000       42.6240       319.400

For illustrative purposes though, storeangles.rct also writes an output called dang-unrot which contains the Euler angles which would be used if not for the possible ambiguity in the angle-fitting step. This doc file also includes some other values that are informative for understanding the outcome of the three Euler angles, for example:

 ;rct/dat   09-OCT-2012 AT 17:49:47   Tilted/dang-unrot.dat
 ;       ANGLE_PSI    ANGLE_THETA    ANGLE_PHI      MIC_NUM       INPLANE    TILT_UNTILTED  MIRROR_FLAG
    1 7  -104.505       45.3470       182.543       1999.00       279.950      -102.493       1.00000
    2 7  -104.505       45.3470       17.2526       1999.00       85.2400      -102.493       1.00000
    3 7  -104.505       45.3470       15.2826       1999.00       87.2100      -102.493       1.00000
    4 7  -104.505       45.3470       178.413       1999.00       284.080      -102.493       1.00000
    5 7  -104.505       45.3470       201.703       1999.00       260.790      -102.493       1.00000
    6 7  -104.505       45.3470       139.353       1999.00       323.140      -102.493       1.00000
[...]
 1581 7  -103.131       42.6240       317.404       2047.00       140.990      -98.3941       1.00000
 1582 7  -103.131       42.6240       259.634       2047.00       198.760      -98.3941       1.00000
 1583 7  -103.131       42.6240       326.614       2047.00       131.780      -98.3941       1.00000
 1584 7  -103.131       42.6240       305.724       2047.00       152.670      -98.3941       1.00000
 1585 7  -103.131       42.6240       81.0241       2047.00       17.3700      -98.3941       1.00000
 1586 7  -103.131       42.6240       57.7942       2047.00       40.6000      -98.3941       1.00000

If looking for mirrored images was enabled in the last iteration of multirefalign.rct (or whichever iteration's alignment doc file was used in storeangles.rct), all three Euler angles will be modified in storeangles.rct.
NOTE: Ths mirroring option was vetted only for phantom data, and not for all combinations of options.

In most cases, this latter file dang-unrot should be equivalent for the reconstruction step, bpclass.rct, provided the tilted-specimen images used in centertilt.rct were Tilted/Unrot/tilt***** instead of Tilted/Winrot/tilt*****. This can be an experiment left for the student.

Some more background information about Euler angles.

Source: eulers.htm     Page updated: 2014/03/05     Tapu Shaikh